TIFR ECE 2017 | Question: 13

Question

Let $A$ be an $n \times n$ matrix. Consider the following statements.

1. $A$ can have full-rank even if there exists two vectors $v_{1} \neq v_{2}$ such that $A v_{1}=A v_{2}$.
2. $A$ can be similar to the identity matrix, when $A$ is not the identity matrix. Recall that two matrices $B$ and $C$ are said to be similar if $B=S^{-1} C S$ for some matrix $S$.
3. If $\lambda$ is an eigenvalue of $A$, then $\exists$ a vector $x \neq 0$ such that $(A-\lambda I) x=0$.

Which of the above statements is/are $\text{TRUE?}$ Choose from the following options.

• A. Only $\text{(i)}$
• B. Only $\text{(ii)}$
• C. Only $\text{(iii)}$
• D. $\text{(i), (ii),}$ and $\text{(iii)}$
• E. None of the above